# Engineering Tables/Properties of Derivatives

Derivative Conditions
1 ${\frac {d}{dx}}c=0$ 2 ${\frac {d}{dx}}(cx)=c$ 3
Elementary Power Rule
${\frac {d}{dx}}(x^{n})=nx^{n-1}$ 4
Sum Rule
${\frac {d}{dx}}\left(f\pm g\pm h\pm \cdots \right)={\frac {df}{dx}}\pm {\frac {dg}{dx}}\pm {\frac {dh}{dx}}\pm \cdots$ 5
Constant Multiple Rule
${\frac {d}{dx}}(cf)=c{\frac {df}{dx}}$ 6
Product Rule
${\frac {d}{dx}}(fg)=f{\frac {dg}{dx}}+g{\frac {df}{dx}}$ 6
Product Rule (Extended)
${\frac {d}{dx}}(fgh)=fg{\frac {dh}{dx}}+fh{\frac {dg}{dx}}+gh{\frac {df}{dx}}$ 7
Quotient Rule
${\frac {d}{dx}}\left({\frac {f}{g}}\right)={\frac {g{\frac {df}{dx}}-f{\frac {dg}{dx}}}{g^{2}}}$ $g\neq 0\,$ 8
Chain Rule
${\frac {df}{dx}}={\frac {df}{dg}}\cdot {\frac {dg}{dx}}$ 9 ${\frac {d}{dx}}\left(f^{n}\right)=nf^{n-1}{\frac {df}{dx}}$ 10
Reciprocal Rule
${\frac {d}{dx}}\left({\frac {1}{f}}\right)=-{\frac {1}{f^{2}}}{\frac {df}{dx}}$ $f\neq 0\,$ 11
Functional Power Rule
${\frac {d}{dx}}\left(f^{g}\right)={\frac {d}{dx}}\left(e^{g\ln f}\right)=f^{g}\left({\frac {g}{f}}\cdot {\frac {df}{dx}}+{\frac {dg}{dx}}\ln f\right)$ $f>0\,$ 12
Logarithm Rule
${\frac {df}{dx}}=f{\frac {d}{dx}}\left(\ln f\right)$ $f>0\,$ 13
Inverse Function Rule
${\frac {df}{dx}}={\frac {1}{\frac {dx}{df}}}$ ${\frac {dx}{df}}\neq 0\,$ Notes:
1. f, g, h are functions of x
2. c, n are constants.